Classifying A-field and B-field configurations in the presence of D-branes - Part II: Stacks of D-branes

TL;DR

This paper extends the classification of B-field and A-field configurations in type II superstring theory to stacks of D-branes with non-abelian gauge theories, integrating gerbe geometry with higher-rank bundles.

Contribution

It provides a detailed analysis of the geometric structures of gauge fields on stacks of D-branes, including non-abelian cases and twisted bundles, building on previous work for single D-branes.

Findings

Classification of A-field as connections on gauge bundles.

Relation between non-integral Chern classes and twisted bundles.

Geometric interpretation of Wilson loops in various gauge theories.

Abstract

In the paper arXiv:0810.4291 we have shown, in the context of type II superstring theory, the classification of the allowed B-field and A-field configurations in the presence of anomaly-free D-branes, the mathematical framework being provided by the geometry of gerbes. Here we complete the discussion considering in detail the case of a stack of D-branes, carrying a non-abelian gauge theory, which was just sketched in the previous paper. In this case we have to mix the geometry of abelian gerbes, describing the B-field, with the one of higher-rank bundles, ordinary or twisted. We describe in detail the various cases that arise according to such a classification, as we did for a single D-brane, showing under which hypoteses the A-field turns out to be a connection on a canonical gauge bundle. We also generalize to the non-abelian setting the discussion about "gauge bundles with non integral Chern classes", relating them to twisted bundles with connection. Finally, we analyze the geometrical nature of the Wilson loop for each kind of gauge theory on a D-brane or stack of D-branes.